Shear modulations and icosahedral twins

Transcription

1 We Microsc. Microanal. Microstruct. 1 (1990) 503 OCTOBER/DECEMBER 1990, PAGE 503 Classification Physics Abstracts J 61.70N Shear modulations and icosahedral twins Michel Duneau Centre de Physique Théorique, Ecole Polytechnique, Palaiseau Cedex, France (Received October 01, 1990; accepted January 15, 1991) Résumé Nous montrons que si deux réseaux La et Lb sont liés par un cisaillement, il existe une transformation displacive simple, correspondant à une modulation périodique de La, qui transforme La en Lb. Nous considérons les maclages icosaèdriques récemment observés dans les alliages AlFeMnSi et montrons qu ils sont liés par des rotations équivalentes au produit de seulement deux tels cisaillements Abstract. show that given two lattices La and Lb, related be a shear transformation, there exists a simple displacive transformation, corresponding to a periodic modulation of La, which transforms La into Lb. We consider recently observed icosahedral twinnings in AlFeMnSi alloys and show that they are related by rotations equivalent to the product of only two such shear transformations. 1. Introduction. Shear transformations are in some the most simple transformations between different lattices. From a geometrical point of a view they are characterized by a direction, an invariant plane and a magnitude. Only in specific situations, such as the martinsitic transformation, do these transformations correspond to real displacements of atoms. In other situations, they are involved in purely geometrical relations between two lattices related by a transition, whereas the actual transformation is believed to be displacive. This is the case of the "second shear" in the martinsitic transformation. In this paper we show that if two lattices are related by a macroscopic linear shear transformation, a simple displacive transformation between these lattices can be devised. This bounded transformation involves a periodic displacement field of small amplitude in such a way that the second lattice turns out to be a modulation of the first one. Such a situation is however restricted to particular pairs of lattices. Generally, two lattices of equal density are related by the combination of 4 successive shear transformations. In this case a displacive transformation can still be performed but the corresponding modulation is far more complex and the displacement field is usually quasiperiodic. Article available at or

2 504 We dont claim that actual displacements are exactly those given those by our construction. Relaxation processes normally would change the trajectories of atoms. But the proposed solutions for displacive transformations are in some sense the most simple and could be associated to simple soft modes. Even if a structural transformation is not observed, the relative orientations of grains in a polycrystalline system can be analysed along the same lines. In the case of twinnings, a generic rotation similarly corresponds to 4 different shears. Of specific interest are the rotations for which only two shear transformations monitor an equivalent displacive transformation. In other words, particular orientational relationships are pointed out by a requirement of simplicity of an actual or potential transformation. In this respect we show that the recently observed icosahedral twins in AlFeMnSi alloys are associated to simple displacive transformations observed icosahedral twins in AlFeMnSi alloys are associated to simple displacive transformations involving only two successive simple modulations. 2. Shear transformation and shear modulation. In this section we show that shear transformations, which are linear and therefore macroscopic transformations, have simple local and displacive analogs called hereafter shear modulations. Let La AZ3 and Lb B Z3 denote two lattices with respective basis {al, a2, a3} and {bl, b2, b3} given by ai Aei and bi Bei (i 1, 2, 3) where fel, e2, e3} is the standard basis of the 3- dimensional space. A and B are 3 x 3 matrices with non vanishing determinants and if La and Lb have the same density we have det (A) ±det (B). Actually, the matrices A and B are not uniquely defined by the lattices La and Lb and similarly, the a-basis and b-basis are not unique. All other equivalent description involve modular matrices U and V, i.e. with integer entries and determinant equal to ±1, in such a way that La AUZ3 and Lb BVZ3. The corresponding basis are respectively {al, a2, 03B13}, given by ai AUei, and {03B21, 32, 03B23}, given by 3i B Vei ( i 1, 2, 3). We shall first consider pairs of lattices La and Lb related by simple shear transformations. By this we mean that Lb SLa where the linear transformation S has the following property: La and Lb have respective basis {al, a2, 03B13} and f,81, 03B22, 03B23} such that where stand S2 are two parameters giving the amplitude of the shear transformation, the direction of which is given by the lattice direction a3 (33. These particular basis are associated, as mentioned above, to particular modular matrices U and V and it follows from (2.1) that Therefore we have BV AUS and consequently the transition matrix between La and 4 reads

3 The 505 Such a shear transformation obviously preserves density since det (S) 1. The shear transformation has an invariant plane P such that SPP which is generated by a3 and s2al - Sla2. P is not in general a lattice plane of La, but if SI and s2 are both rational numbers. Any lattice point x of La reads x Exs as AUX, where X is a column vector X with integer entries.ci, X2 and X3. In view of (2.1) the mapping S transforms a lattice point x 03A3xi03B1i of La into the lattice point y 03A3xi03B2i BVX of Lb. The coordinates Y of y in the a-basis satisfy y AUY from which we easily derive Obvioulsy the distance between x and y S(x) is not bounded since Now, assuming that La and Lb are related by a such a simple shear transformation, we show that a simple transformation from La to Lb can be performed by means of bounded displacements: this means that there exists a global mapping E from La to Lb, that well call shear modulation for obvious reasons, such that for ail x in L the distance between x and z E(x) in Lb remains bounded. Any lattice point x Exiai of La can be decomposed in the /?-basis: with - Fig. 1. white lattice L, generated by {al, 03B12}, is transformed into L generated by 1 a 1 a2 + s03b11}. The shear transformation S, represented by dotted arrows is not bounded, while the shear modulation E, represented by black arrows, is bounded. So the third coordinate y3 is not, in general, an integer (since x is not a lattice point of Lb). If Rnd(x) denotes the round function, i.e. Rnd(x) n if n-1/2 x n+1/2, then the point

4 506 defined by z x103b21 + x203b22+rnd(x3 - SIX1 - S2X2),33 belongs to the lattice Lb. Furthermore the difference between x and z 03A3(x) is given by Where Frac(x) x-rnd(x) is a function with value between -1/2 and 1/2. So the distance between x and z is always bounded by ~03B13~/2~03B23~/2 (see Fig. 1). It was proved in [2] that the shear modulation E:x --+ z from La to Lb is actually one-to-one and can be extended to a periodic piecewise linear mapping on the whole physical space. E differs from the identiy mapping by a bounded displacement d : where {x1, X2, x3} are the coordinates of x in the a-basis of La. Since the Frac function is periodic with period 1, Frac(x + n) Frac (x), the displacement field d(x) is a periodic modulation field which reads d(x)frac(q.x)03b13 for some q in the reciprocal space. Finally, Lb can be considered as a periodic modulation of La, although the period of d is generally incommensurate with La. Now let La AZ3 and Lb BZ3 be any two lattices with equal density (det A det B). The linear transformation BA-1 from La to Lb is represented by the matrix T A-1 B in the natural basis of La and det T±1. In general, these lattices will not be related by a unique displacive transformation as above, even with the free choice of the modular matrices U and V : there are not enough continuous of shear transforma- parameters. But the transition matrices T can be decomposed into a product tions. More precisely, it is proved in [2] that there exists a modular matrix U, with integer entries and determinant ±1, and at most 4 shear transformations So, Sl, S2 and S3 such that (compare to (2.3)) where Usually, the modular matrix is the identity matrix but for particular transition matrices, a change of basis, performed by U, is necessary. Once the transition matrix T is decomposed into a product of shear transformations we can apply the above results to build a bounded one to one mapping E between La and Lb. This is achieved simply by replacing each linear shear transformation Si, associated to the matrix Si, by the corresponding bounded shear modulation E; :03A3 EoE3E2Ei. The resulting displacement field d(x) is defined by 03A3(x) x + d(x). However this modulation is far more complicated: in particular d involves as many frequencies {qo, q3, q2, q1} as shears transformations in the decomposition. As a consequence the modulation is generally quasiperiodic.

5 The 507 The decomposition (2.11) can be obtained by an explicit algorithm. For simplicity, we only give here the 2-dimensional analog of this procedure: if T is a 2 x 2 matrix with determinant ±1, there exist a 2 x 2 modular matrix U and at most 3 shear transformations So, SI and S2 such that where Actually the product of shear matrices reads The three parameters s0,s1 and s2 are easily derived from the entries of T. Notice that a modular matrix U, different from the identity, is needed if T2i 0; actually, if U I and T2l 0, we get 82 0 which would suppose that Tll T22 1. If La and Lb denote the two square lattices related by a 7r/4 rotation, T is simply the 7r/4 rotation and the above decomposition reads Fig quasiperiodic one to one mapping between the two square lattices at 7r/4. If x xlal + x2a2 is in La and y ylbl + Y2b2 is in 4 the corresponding bounded one to one mapping from La to Lb (see Fig. 2) is given by Remark Such bounded one to one mappings instance the case of two square lattices La and 4 in the plane between lattices are not obvious. Consider for with a 7r/4 relative orientation. A

6 508 natural candidate for such a map consists in mapping a point x of La onto the origin y of the square of Lb which contains x. However, squares of Lb may contain 0, 1 or 2 points of La so that the one to one property cannot be satisfied. The number of shear transformations, 4 in 3D and 3 in 2D, is a maximum. In 3D a transition matrix T A- B involves 8 independent continuous parameters because det[t]±1. Since each shear transformation involves only 2 parameters, four of them must be combined to cover all cases. But, for certain lattices and transition matrices the decomposition may involves less shear transformations. In view of the decomposition (2.11) these cases correspond to at least one shear transformation being equal to the identity. This reduction is exceptional and we think that it may have an important physical meaning by specifying particular orientational relationships. For instance if Lb R La, where R is a rotation, the transition matrix is T A -1 B A-1 RA and the decomposition reads A -1 RA USoS3S2 SI. The point is that for exceptional rotations R, the decomposition may involve only 3 or even 2 shear transformations. This is actually observed in the case of icosahedral twinnings as we show in next section. 3. Icosahedral twins. In this section we show that icosahedral twins of cubic lattices, i.e. twins related by a 21r/5 rotation can be viewed as around a pseudo 5-fold axis of type 103C40> (r is the golden mean (1 + 5)/2), corresponding to a coincidence lattice, up to simple shear transformations (see also [3]). Let al (1, 0) and a2(1/2, 3/2) denote the generators of the 2D-lattice L corresponding to the (111) -plane of a cubic lattice of convenient parameter. Then where A 1 1/2 and Z2 is the standard square lattice. Let Aa denote the sub-lattice of L spanned by al 2ai + 2a2 and a2 a2. Thus Aa is of index 2 in L and can be obtained as follows: Now we consider a lattice A a which is deduced from Aa by a shear transformation sa which transforms the basis {03B11, 03B12} into {03B1 1, 03B1 2} : a 03B11 and a 03B B12. Consequently Similarly let Ab denote the sub-lattice of L spanned by (31 4ai - a2 and,q2 2aI. Thus Ab is of index 2 in L and can be obtained as follows: A lattice b is derived from Ab by a shear transformation Sb which transforms the basis {(3t, 03B22} into {,6l 03B2 2} : 03B2 1 ( and ( Therefore

7 Notice 509 Notice that a and b have equal density since det(m) det(n) 2. The point is that b can be deduced from a by a mere rotation; this can be checked by computing the following norms and scalar products: 03B1 1 1,3 l 23, 03B1 2 03B2 2 2 and (03B1 1, 03B1 2) (03B2 1, 03B2 2) 603C4-3. More precisely if follows from the above definition that A b AN (SbS-1a) M-1 A-1 a. A simple computation gives the transition matrix which is a rotation r of angle Arccos((303C4-2)/4) Arcsin(03C43/4) ~ Such orientation relations between grains of cubic systems have been observed in rapidly quenched alloys of Al75Mn15-xFexSi10 (see [1] ) with x 5 or 10. As pointed out in [1], this rotation around an [111] axis of a cubic lattice is equivalent to a 203C0/5 rotation around the pseudo "five-fold axis" 103C40>. More precisely, a five-fold rotation around [103C40] generates five different variants of the cubic system. Any two of them share a three-fold axis and they are related by the above rotation. Fig that since 5/3 is a good approximant of r,,ql therefore b is, in some sense, almost a sub-lattice of L. r03b11 is very close to 03B21 4at - a2 and

8 510 Acknowledgements. Part of this work is closely related with previous works with C. Oguey to them for many helpful and constructive discussions. and A. Thalal. 1 am indebted References [1] BENDERSKI L.A., CAHN J.W, GRATIAS D., Phil. Mag. 60 (1989) 837. [2] DUNEAU M., OGUEY C., J. Phys. A 24 (1991) 461. [3] DUNEAU M., OGUEY C., Europhys. Lett. 13 (1990) 67.